Difference between revisions of "2023 AMC 10B Problems/Problem 24"
Technodoggo (talk | contribs) (→Solution 1) |
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import graph; | import graph; | ||
Label f; | Label f; | ||
− | + | size(5cm); | |
unitsize(0.7cm); | unitsize(0.7cm); | ||
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unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
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unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
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unitsize(0.7cm); | unitsize(0.7cm); | ||
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xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
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unitsize(0.7cm); | unitsize(0.7cm); | ||
− | + | size(5cm); | |
xaxis(-5,5,Ticks(f, 5.0, 1.0)); | xaxis(-5,5,Ticks(f, 5.0, 1.0)); | ||
yaxis(-5,5,Ticks(f, 5.0, 1.0)); | yaxis(-5,5,Ticks(f, 5.0, 1.0)); |
Revision as of 15:14, 15 November 2023
What is the perimeter of the boundary of the region consisting of all points which can be expressed as with
,
and
?
Solution 1
Notice that this we are given a parametric form of the region, and
is used in both
and
. We first fix
and
to
, and graph
from
:
Now, when we vary from
to
, this line is translated to the right
units:
We know that any points in the region between the line (or rather segment) and its translation satisfy and
, so we shade in the region:
We can also shift this quadrilateral one unit up, because of . Thus, this is our figure:
The length of the boundary is simply (
can be obtained by Pythagorean theorem, since we have side lengths
and
.). This equals
~Technodoggo